Completing the Square for IGCSE Maths
Rewriting quadratics in the form a(x+p) squared + q to find turning points. This subtopic is part of Algebra in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding comple
What You Need to Know
Rewriting quadratics in the form a(x+p) squared + q to find turning points. This subtopic is part of Algebra in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding completing the square is essential for achieving a strong grade in your IGCSE Maths exam.
Understanding Completing the Square
Completing the square rewrites x² + bx + c in the form (x + p)² + q, where p = b/2 and q = c − (b/2)². This is tested in IGCSE Extended to: find the minimum (or maximum) value of a quadratic expression, find the coordinates of the vertex of a parabola, and solve quadratic equations leaving answers as exact surds. For ax² + bx + c with a ≠ 1, take out factor a first.
Step-by-Step Method
- 1
Check the coefficient of x²
If a = 1, proceed directly. If a ≠ 1, factor out a from the x² and x terms first: ax² + bx + c = a[x² + (b/a)x] + c.
- 2
Halve the x-coefficient
Take the coefficient of x, halve it, then write (x + half)². For x² + 6x, write (x + 3)².
- 3
Subtract the square of the half
(x + 3)² = x² + 6x + 9, so x² + 6x = (x + 3)² − 9. The balance term is −(b/2)².
- 4
Add the constant
Include the original constant c: x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7.
- 5
Read off the minimum/vertex
In (x + p)² + q: the vertex is (−p, q), and q is the minimum value (if a > 0) or maximum value (if a < 0).
Worked Example
Question
(a) Write x² − 8x + 3 in the form (x + p)² + q. (b) Hence state the minimum value of x² − 8x + 3 and the value of x at which it occurs.
Solution
Part (a): x² − 8x + 3 Halve −8: −8/2 = −4, so write (x − 4)². (x − 4)² = x² − 8x + 16 Balance: x² − 8x = (x − 4)² − 16 Add constant: x² − 8x + 3 = (x − 4)² − 16 + 3 = (x − 4)² − 13 Part (b): (x − 4)² ≥ 0 always. Minimum value = −13, occurring when x − 4 = 0, i.e. x = 4. Answers: (a) (x−4)² − 13 (b) Minimum = −13 at x = 4
Exam Tips for Completing the Square
- Halve the x-coefficient to find p — never use the full coefficient.
- After writing (x+p)², subtract (b/2)² to maintain equivalence — forgetting this is the most common error.
- The minimum of (x+p)²+q is q (not p), occurring at x = −p.
- For solving, completing the square always gives exact surd answers — use it when the question says 'exact form' or 'give in the form a+b√c'.
Practice Questions
Q1: Write x² + 10x − 4 in completed square form. State the minimum value.
Show hint
Half of 10 = 5. So (x+5)² − 25 − 4 = (x+5)² − 29. Minimum = −29.
Q2: Solve x² − 6x + 2 = 0 by completing the square, giving exact answers.
Show hint
(x−3)² − 7 = 0 → (x−3)² = 7 → x = 3 ± √7.
Q3: Write 2x² − 12x + 5 in the form a(x+p)² + q.
Show hint
Factor out 2: 2[x² − 6x] + 5. Complete: 2[(x−3)² − 9] + 5 = 2(x−3)² − 18 + 5.
Frequently Asked Questions
What is completing the square in IGCSE Maths?
Rewriting quadratics in the form a(x+p) squared + q to find turning points.
Is completing the square in the Core or Extended syllabus?
Completing the Square is part of the Extended only syllabus for IGCSE Mathematics 0580.
How do I revise completing the square effectively?
Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on completing the square rather than trying to cover everything at once.
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